Finding the Inverse of a Linear Transformation

[UrbanXrisis]

how would one find the inverse of the linear transformation:

$$y_1=4x_1-5x_2$$
$$y_2=-3x_1+4x_2$$

this was never taught in class, could someone give a little advice as how I would do this?

I know the answer has to be in the form of

$$x_1=ay_1+by_2$$
$$x_2=cy_1+dy_2$$

could someone explain this process?

 

[NateTG]

You're solving the system of equations for $x_1$ and $x_2$.


One way to do it would be to solve the first equation for $x_1$ and then substitute into the second equation.

Another method would be to add the equations together using suitable coefficitents so that one of the $x$'s is eliminated, and then solve for the other.

In principle, this should be no different than dealing with, for example:
$$9=4x_1-5x_2$$
$$7=-3x_1+4x_2$$

 

[neutrino]

Yet another way is to write the equations in matrix form. (Left) Multiply both sides by the inverse of the coefficient matrix.